Secant Matrices Research Articles

Implicit to Explicit Algorithm for ABAQUS Standard User-Subroutine UMAT for a 3D Hashin-Based Orthotropic Damage Model

This study examines a new approach to facilitate the convergence of upcoming user-subroutines UMAT when the secant material matrix is applied rather than the conventional tangent (also known as Jacobian) material matrix. This algorithm makes use of the viscous regularization technique to stabilize the numerical solution of softening material models. The Newton–Raphson algorithm predictor-corrector of ABAQUS then applies this type of viscous regularization to a UMAT using only the secant matrix. When the time step is smaller than the viscosity parameter, this type of regularization may be unsuitable for a predictor-corrector with the secant matrix because its implicit convergence is incorrect, transforming the algorithm into an undesirable explicit version that may cause convergence problems. A novel 3D orthotropic damage model with residual stresses is proposed for this study, and it is analyzed using a new algorithm. The method’s convergence is tested using the proposed implicit-to-explicit secant matrix as well as the traditional implicit and explicit secant matrices. Furthermore, all numerical models are compared to experimental data. It was concluded that both the new 3D orthotropic damage model and the new proposed time step algorithm were stable and robust.

Variable order secant matrix technique applied to the elastic postbuckling of arches

AbstractThis paper presents the implementation of Reissner's (J. Appl. Math. Phys. 1972; 23:795–804) strain–displacement relations for a planar beam using a variable order secant matrix (VOSM) technique. The secant matrices are formed using a Taylor expansion of the strain energy and hence the order of the secant and tangent matrices to be included in the nonlinear analysis can be varied depending upon the degree of nonlinearity involved in the problem. Nevertheless, a maximum of 24 matrices is sufficient to obtain predicted internal stress resultants to within 0.2% of the converged solution in all the problems tested. Using a three‐noded isoparametric beam element, the accuracy of the VOSM method is demonstrated for the elastic postbuckling of arches. Copyright © 2007 John Wiley & Sons, Ltd.

Marguerre shell type secant matrices for the postbuckling analysis of thin, shallow composite shells

The postbuckling behaviour of thin shells has fascinated researchers because the theoretical prediction and their experimental verification are often different. In reality, shell panels possess small imperfections and these can cause large reduction in static buckling strength. This is more relevant in thin laminated composite shells. To study the postbuckling behaviour of thin, imperfect laminated composite shells using finite elements, explicit incremental or secant matrices have been presented in this paper. These incremental matrices which are derived using Marguerre's shallow shell theory can be used in combination with any thin plate/shell finite element (Classical Laminated Plate Theory - CLPT) and can be easily extended to the First Order Shear deformation Theory (FOST). The advantage of the present formulation is that it involves no numerical approximation in forming total potential energy of the shell during large deformations as opposed to earlier approximate formulations published in the literature. The initial imperfection in shells could be modeled by simply adjusting the ordinate of the shell forms. The present formulation is very easy to implement in any existing finite element codes. The secant matrices presented in this paper are shown to be very accurate in tracing the postbuckling behaviour of thin isotropic and laminated composite shells with general initial imperfections.

Improved Secant Matrices for the Postbuckling Analysis of Thin Composite Plates

To study the postbuckling behavior of imperfect laminated composite plates, improved incremental or secant matrices are presented in this paper using what is called additional displacement formulation (ADF). These secant matrices are derived using the Marguerre's shell theory and they can be used in combination with any thin plate finite element. The advantage of the present formulation is that it involves no numerical approximation in forming the initial imperfection matrices as opposed to earlier secant matrices published in the literature using total displacement formulation. With the addition of shear stiffness matrix and little modification, the present incremental matrices could be extended to model postbuckling behavior of plates using the first-order shear deformation theory. The secant matrices presented in this study are shown to be very accurate in tracing the postbuckling behavior of thin isotropic and laminated composite plates with general initial imperfections.

Explicit incremental matrices for the postbuckling analysis of thin plates with small initial curvature

The postbuckling behaviour of thin plates is an important phenomenon in the design of thin plated structures. In reality plates possess small imperfections and the behaviour of such imperfect plates is of great interest. To numerically study the postbuckling behaviour of imperfect plates explicit incremental or secant matrices have been presented in this paper. These matrices can be used in combination with any thin plate element. The secant matrices are shown to be very accurate in tracing the postbuckling behaviour of thin plates.